Chapter 10
Non-equilibrium physics
Learning goals
•
•
•
•
Why are non-equilibrium aspects important to cold atoms?
What is special about isolated systems?
What is a collapse and revival phenomenon?
What is a bath and how can it be modeled?
10.1
Overview
In (quantum) statistical mechanics we are usually given a Hamiltonian Ĥ that should describe
the system and we are faced with the problem to calculate expectation values relevant for a
given experimental measurement. Generically we assume to be either in the ground state | 0 i
or in a thermal state ⇢ˆ = e Ĥ , where 1/ = kB T . In both cases the expectation values are
given by
h
i
h
i
hÂ(t)i = Tr Â(t)e Ĥ
or
hÂ(t)B̂(t0 )i = Tr Â(t)B̂(t0 )e Ĥ .
(10.1)
These simple formulas are somewhat deceiving. Either we need to be strictly in the groundstate ( ! 1), or we assume that the system reached a thermal equilibrium with some “large”
bath that provided the notion of a temperature T . Neither of these assumptions are generically
fulfilled in experiments with ultra-cold atoms.1
In the present chapter we are looking into two di↵erent situations where one can capitalize on
this specific properties of quantum gases. First, if there is no good bath attached which can
thermalize our system, we have the unique opportunity to observe pristine quantum many-body
dynamics. We can track the coherent evolution of some complicated many-body state and
observe its intrinsic dynamics. In a solid state system such a coherent quantum dynamics is
often contaminated by the influence of other, unwanted, degrees of freedom.
In a second example we are studying how one can re-introduce a custom-tailored bath. Given
the high degree of control on the Hamiltonian of the system, one can try to extend this control to
the non-unitary evolution of an open system. Indeed we will see how using the specific tailoring
of a bath, one can prepare specific ground states of model Hamiltonians.
The present discussion of non-equilibrium phenomena in general and their appearance in quantum gas experiment is far from being exhaustive. The purpose of this chapter is to give a flavor
of the physics in a non-equilibrium setup using two simple examples.
1
Which raises the questions why we call them cold...?
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10.2
Isolated systems: pure quantum dynamics
In a closed quantum systems the dynamics of many-body system can be observed without the
contamination from other degrees of freedom. This is particularly important for the observation
of dynamical properties that involve excited states. The case of repulsively bound states is a
prime example.
10.2.1
Repulsively bound pairs
Let us consider a one-dimensional Bose Hubbard model
H=
t
X
â†i âi±1 +
i
UX
n̂i (n̂i
2
1).
(10.2)
i
For the discussion of repulsively bound pairs we are interested in the two-particle problem. To
this end, we want to switch back to a first quantized description as we restricted to a specific
particle number. We are writing for the two particle wave function (x, y), where x (y) is the
discrete coordinate of the first (second) particle, respectively. The Schrödinger equation now
reads
h
i
˜
˜
H (x, y) =
t x t y + U x,y (x, y) = E (x, y).
(10.3)
The first two terms are the discrete Laplace operators
˜ x (x, y) = (x + 1, y) + (x 1, y)
˜ y (x, y) = (x, y + 1) + (x, y 1)
2 (x, y),
(10.4)
2 (x, y)
(10.5)
and the last term encodes that the particles only interact when they are on the same site. The
Schrödinger equation does not depend on the center of mass R = (x + y)/2, so we write
(x, y) = eiKR
K (r)
with
r=x
Inserted into the the Schrödinger equation we obtain
h
i
2t ˜ r,K + EK + U r,0 K (r) = E
where
˜ r,K
K (r)
= cos (K/2) [
K (r
+ 1) +
K (r
y.
(10.6)
K (r),
1)
2
(10.7)
K (r)] ,
(10.8)
and the center of mass kinetic energy is given by
EK = 4t[1
cos(K/2)].
The free (U = 0) part is solved with a Fourier-transformation K (k) = p1N
which yields
⇢
4t cos(K/2)[1 cos(k)] +EK K (k) = E K (k).
|
{z
}
(10.9)
P
r
exp(irk)
K (r)
(10.10)
✏K,k
We are now ready to tackle the interacting problem.
Lippmann-Schwinger equation
We cast equation (10.3) into the following form
h
i
2t ˜ r,K EK + E K (r) = U r,0 K (r).
|
{z
}
“source term”
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(10.11)
In the standard way we solve for the Greens function where we replace the source term with a
simple point source
h
i
2t ˜ K,r EK + E G(r, E, K) = r,0 .
(10.12)
After a Fourier-transform (r ! k) and making use of (10.10) we find
G(k, E, K) =
E
1
.
(EK + ✏K,k ) + i⌘
(10.13)
Equation (10.11) can now be solved via a convolution of the actual source [U r,0 K (r)] with the
Greens function. As usual we can add a solution of the “free” part which is here just exp(ikr).
With this we arrive at the Lippmann-Schwinger equation
Z
ikr
+ dr0 G(r r0 , E, K) r0 ,0 U K (r0 ),
(10.14)
K (r) = e
R dk
with G(r, E, K) = 2⇡
G(k, E, K)e
above equation reduces to
ikr .
K (r)
Owing to the delta function in the source term the
= eikr + G(r, E, K)U
This is still a non-trivial equation for the wave function
find
K (0)
= 1 + G(0, E, K)U ]
K (0)
)
K (0)
K (0).
K (r).
=
1
(10.15)
However, we can set r = 0 to
1
.
U G(r = 0, E, K)
(10.16)
Putting everything together we find for the full solution
scattering
K
(k) =
k,k0
+ G(k 0 , E, K)
1
U
.
U G(r = 0, E, K)
(10.17)
The first part describes an incoming wave at k 0 . The second part is the scattering part int
momentum k. In addition to the scattered wave, there is a resonance at 1 U G(r = 0, E, K) = 0,
where we can neglect the first part. Such a solution which does not depend on the incoming
wave is called a bound state. The question is now for which energy E and total momentum
K does such a bound state exist. Performing the Fourier transform on the Greens function we
arrive at
p
Ebs (K) = 4t + sign(U ) U 2 + 8t2 (1 + cos(K)).
(10.18)
In one dimension, this energy always lies above (below) the two-particle continuum ✏K,k for
repulsive (attractive) interaction, respectively. Hence, this bound state is stable. For attractive
U < 0, this comes as no surprise. What is however, the physics of such a bound state for
repulsive interactions?
In the presence of a band gap, i.e., on a lattice, the repulsively bound state is stable. If we
would like to take a two-particle tower apart, we would have to accommodate the excess energy
U into kinetic energy. However, this energy might lie in the band gap. It turns out, that in one
dimension, the additional constraint in the preservation of total momentum makes this energy
always to lie in the band gap!
Such a repulsively bound pair can only survive if nobody else can carry away the excess energy
U . The fact that this bound states have been observed by Winkler et al. in Nature 853, 441
(2006) can be seen as a clear sign of the perfect isolation of cold atomic gases.
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10.2.2
Collapse and revival
Another example of such non-equilibrium dynamics arises after a quench (sudden change) from
a superfluid deep into a Mott insulator (e.g. to t = 0). The super fluid order parameter = hâi
for the ground state at t = 0 is clearly zero. What happens to the superfluid order parameter
(⌧ ) as a function of time ⌧ after the quench?
P
To answer this question we note that the eigenstates of the Hamiltonian H = U2 i n̂(n̂ 1) are
simply given by Fock states
H|ni = U n(n 1)|ni.
(10.19)
In order to track the time evolution after the quench we only need to expand the initial ground
state in terms of the new eigenstates
X
(⌧ = 0) = SF =
↵n1 ,n2 ,... |n1 , n2 , . . .i.
(10.20)
n1 ,n2 ,...
The time evolution is now given by
X
(⌧ ) =
↵n1 ,n2 ,... ei⌧ U [n1 (n1
1)+n2 (n2 1)+··· ]
n1 ,n2 ,...
|n1 , n2 , . . .i.
(10.21)
From the above equation we immediately see that after time ⌧ = 2⇡
U the exponential only
contains multiples of 2⇡ and hence the original wave function is restored! Consequently, also
the superfluid order parameter will be restored to its original value. Such collapse and revival
experiments have been reported in Nature 419, 51 (2002).
10.3
Driven-dissipative systems
A very di↵erent non-equilibrium aspect of cold atoms can be revealed by the engineering of a
suitable bath for quantum gases. In this approach, the absence of a bath and the high level of
control on the dynamics of trapped quantum gases can be combined in the design of a suitable
artificial bath. In order to understand this we need to be able to describe a system weakly
coupled to a bath.
10.3.1
The quantum master equation
Let us start from a system weakly coupled to a bath as described by the Hamiltonian
H = Hsys + Hint + Hbath .
(10.22)
We are now targeting an equation of motion for our system density matrix ⇢sys where we don’t
want to keep track of what is happening to the bath degrees of freedom. Under the following
three condition one can derive the quantum master equation
1. The density matrix of the full system at time t = 0 can be written as ⇢(0) = ⇢sys ⌦ ⇢bath .
In other words, the system and the bath are initially not entangled.
2. @t ⇢bath = 0: The state of the bath is not changed by what happens in our system.
3. The time dependent correlation function of the bath are not changed by the coupling to
the system, i.e., the bath is much faster than the system.
Using these assumptions and considering an interaction Hamiltonian of the form
X
Hint =
Jˆ↵† Ô↵bath + H.c
↵
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(10.23)
where Jˆ↵ denotes a system operator we arrive at the master equation
@t ⇢sys (t) =
i[Hsys , ⇢sys (t)]
i
Xh
ˆ† ˆ
ˆ ˆ†
i
↵ J↵ J↵ + ✏↵ J↵ J↵ , ⇢sys (t)
↵
+
X
↵
+
X
✓
K↵ Jˆ↵ ⇢sys (t)Jˆ↵†
✓
G↵ Jˆ↵† ⇢sys (t)Jˆ↵
↵
(10.24)
o◆
1 n ˆ† ˆ
J↵ J↵ , ⇢sys (t)
2
o◆
1 n ˆ ˆ†
J J , ⇢sys (t)
.
2 ↵ ↵
The first three terms describe the coherent system evolution, the Lamb- and Stark-shift respectively. The last two terms account for the non-unitary evolution induced by the coupling to the
external bath.
Here we tried to argue that quantum gases are generically not connected to a bath. What is
now the value of the above equation for our discussion? It turns out, that as much as one can
design system Hamiltonians, one can also try to design suitable “jump operators” J↵ in order
to directly implement a desired non-unitary evolution. This is particularly interesting if the
long-time limit of such a designed non-unitary evolution is not some generic mixed state but
a well defined pure quantum state. In such a case, one can use a bath to prepare a desired
(ground) state. We illustrate this procedure on the example of a recent publication in Nature
Phys. 878, 4 (2008).
10.3.2
Bath tailoring
We can write directly a desired Liouvillian
@t ⇢sys (t) =
i[Hsys , ⇢sys (t)] +
X
K↵
↵
✓
Jˆ↵ ⇢sys (t)Jˆ↵†
o◆
1 n ˆ† ˆ
J J , ⇢sys (t)
.
2 ↵ ↵
(10.25)
This generically leads to a mixed steady state [⇢sss = ⇢sys (t ! 1)]. However, under special
circumstances, the long time limit can be a pure state |Di. This is the case if
ˆ
1. |Di is a an eigenstate of all jump operators J†|Di
= 0 with eigenvalue 0, and
2. |Di is “compatible” with the unitary dynamics, i.e., H|Di = E|Di.
In the language of quantum optics |Di is called a “dark state”.
Long-range order by local dissipation
How can we now design a set of jump operators that lead to such a dark state. Consider a
Bose-Hubbard model where we set the interaction U to zero. The ground state |BECi is given
P
by all particles residing in the k = 0 mode â†k=0 / i â†i . Hence, if we want to prepare the
state |BECi via a designed bath, we need to make sure that only states survive where the phase
between adjacent sites is set to zero. The following jump operators would do the job
Jˆ↵ = Jˆi = (â†i + â†i+1 )(ai
ai+1 ).
(10.26)
Clearly J↵ |BECi = 0 for all ↵. Moreover, the state |BECi is compatible with the unitary
dynamics (at U = 0). How can we now design such a jump operator in an experiment?
Imagine a cubic optical lattice where at each link of the original (system) lattice there is another
(high-energy) state available, cf. Fig. 10.1. If we now pump the system with Rabi frequencies
±⌦ from the lower to the upper “band”, the system framed in red constitutes a standard ⇤system. If the particle is allowed to decay back down to the lower band, we indeed deal with a
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Figure 10.1: Setup for dissipative cooling.
jump operator of the form C↵,i,j â†↵ (âi âj ). Two problems need to be addressed: (i) How can
the particle fall back down to the lower sites. (ii) How are the coefficients C↵,i,j related to the
desired jump operator in Eq. (10.26).
The first problem can be solved by immersing the whole setup into a big (not necessarily T = 0)
Bose Einstein Condensate. The particles falling down can then transfer their excess energy to
a Bogoliubov excitation of the “big” condensate. The details of the coupling then also encode
the properties of the coefficients C↵,i,j . However, the details are not crucial, as long as the local
antisymmetric combination is pumped up, and hence |BECi is a dark state. More details about
this setup are discussed in Nature Phys. 878, 4 (2008).
What was the magic of this bath tailoring? The crucial aspect was that we found a local pump,
where antisymmetric phase configurations on a lattice are removed by the jump operator. The
resulting dark state, however, had a long-range phase coherence. Hence, via local operators we
managed to prepare a long-range correlated state. Following the aforementioned publication,
various proposals were formulated where the long-range correlated state is much more complicated than the |BECi discussed here. This includes d-wave superconducting fermion systems or
topological p-wave superconductors.
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